论文标题
圆柱形对称卷曲 - 卷曲问题和具有奇异电位的相关schrödinger方程的多种解决方案
Multiple solutions to cylindrically symmetric curl-curl problems and related Schrödinger equations with singular potentials
论文作者
论文摘要
我们寻找多个解决方案$ \ mathbf {u} \ colon \ colon \ mathbb {r}^3 \ to \ mathbb {r}^3 $ to curl-curl问题\ [\ nabla \ nabla \ times \ nabla \ nabla \ nabla \ nabla \ nabla \ nabla \ times \ times \ times \ mathbf {u} = x \ in \ mathbb {r}^3，\ \]，带有非线性函数$ h \ colon \ colon \ mathbb {r}^3 \ times \ times \ mathbb {r}^3 \ to \ mathbb {r {r}^3 $，在$中至关重要。 $ h（x，x，\ mathbf {u}）= | \ mathbf {u} |^4 \ mathbf {u} $，或在Infinity时具有亚临界增长。如果$ h $在$ \ mathbf {u} $和$ a = 1 $以下是径向，那么我们表明上述问题的解决方案与以下schrödinger方程\ [-Δu+\ frac+\ frac {a} a} a} {a} {r^2} {r^2} u = f（x，x，u），\ quad \ clon \ clon \ qudy \ clon \ clon \ qualon clon \ clon \ qualon \ clon \ qudy是一对一。 \ Mathbb {r}，\]其中$ x =（y，z）\ in \ mathbb {r}^2 \ times \ times \ mathbb {r} $，$ r = | y | $和$ a \ ge 0 $。在关键情况下，仅在自主情况$ a = 0 $的情况下研究了后一个方程的多样性问题，并且由于缺乏共生不变性，可用的方法似乎不足以解决涉及单一潜力的问题，即$ a \ neq 0 $。因此，我们为关键的卷曲卷曲问题开发方法，并显示两个方程式的多重状态。相反，在亚临界情况下，研究schrödinger方程在较高维度上，我们发现这两个问题都无限地结合了许多界面状态。
We look for multiple solutions $\mathbf{U}\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem \[ \nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3, \] with a nonlinear function $h\colon\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3$ which is critical in $\mathbb{R}^3$, i.e., $h(x,\mathbf{U})=|\mathbf{U}|^4\mathbf{U}$, or has subcritical growth at infinity. If $h$ is radial in $\mathbf{U}$ and $a=1$ below, then we show that the solutions to the problem above are in one-to-one correspondence with the solutions to the following Schrödinger equation \[ -Δu+\frac{a}{r^2}u=f(x,u),\qquad u\colon\mathbb{R}^3\to \mathbb{R}, \] where $x=(y,z)\in \mathbb{R}^2\times \mathbb{R}$, $r=|y|$ and $a \ge 0$. In the critical case, the multiplicity problem for the latter equation has been studied only in the autonomous case $a=0$ and the available methods seem to be insufficient for the problem involving the singular potential, i.e., $a\neq 0$, due to the lack of conformal invariance. Therefore we develop methods for the critical curl-curl problem and show the multiplicity of bound states for both equations. In the subcritical case, instead, studying the Schrödinger equation in higher dimensions, we find infinitely many bound states for both problems.
